If you exceed your iCloud storage limit

When you go over your iCloud storage limit, your messages and attachments don't upload to iCloud, and your devices won't back up or download to iCloud. You can still access your most recent messages, but they won't be stored in iCloud and deleted messages will not update on all of your devices. To keep using Messages in iCloud, upgrade your storage or free up space by deleting files that you don't need.

If messages aren't updating across your devices

When you turn on Messages in iCloud for the first time, your messages sync chronologically, starting with the newest. That means older messages might take longer to appear.

Make sure that you turn on Messages on all of your devices, and that your device is plugged into a power outlet and connected to a network. After your messages sync, new messages won't require a power connection to stay updated.